Metamath Proof Explorer

Theorem bnj1371

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1371.1 
|- B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
bnj1371.2 
|- Y = <. x , ( f |` _pred ( x , A , R ) ) >.
bnj1371.3 
|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1371.4 
|- ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
bnj1371.5 
|- D = { x e. A | -. E. f ta }
bnj1371.6 
|- ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
bnj1371.7 
|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1371.8 
|- ( ta' <-> [. y / x ]. ta )
bnj1371.9 
|- H = { f | E. y e. _pred ( x , A , R ) ta' }
bnj1371.10 
|- P = U. H
bnj1371.11 
|- ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
Assertion bnj1371 
|- ( f e. H -> Fun f )

Proof

Step Hyp Ref Expression
1 bnj1371.1 
 |-  B = { d | ( d C_ A /\ A. x e. d _pred ( x , A , R ) C_ d ) }
2 bnj1371.2 
 |-  Y = <. x , ( f |` _pred ( x , A , R ) ) >.
3 bnj1371.3 
 |-  C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4 bnj1371.4 
 |-  ( ta <-> ( f e. C /\ dom f = ( { x } u. _trCl ( x , A , R ) ) ) )
5 bnj1371.5 
 |-  D = { x e. A | -. E. f ta }
6 bnj1371.6 
 |-  ( ps <-> ( R _FrSe A /\ D =/= (/) ) )
7 bnj1371.7 
 |-  ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
8 bnj1371.8 
 |-  ( ta' <-> [. y / x ]. ta )
9 bnj1371.9 
 |-  H = { f | E. y e. _pred ( x , A , R ) ta' }
10 bnj1371.10 
 |-  P = U. H
11 bnj1371.11 
 |-  ( ta' <-> ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
12 9 bnj1436 
 |-  ( f e. H -> E. y e. _pred ( x , A , R ) ta' )
13 rexex 
 |-  ( E. y e. _pred ( x , A , R ) ta' -> E. y ta' )
14 12 13 syl 
 |-  ( f e. H -> E. y ta' )
15 11 exbii 
 |-  ( E. y ta' <-> E. y ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
16 14 15 sylib 
 |-  ( f e. H -> E. y ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) )
17 exsimpl 
 |-  ( E. y ( f e. C /\ dom f = ( { y } u. _trCl ( y , A , R ) ) ) -> E. y f e. C )
18 16 17 syl 
 |-  ( f e. H -> E. y f e. C )
19 3 abeq2i 
 |-  ( f e. C <-> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
20 19 bnj1238 
 |-  ( f e. C -> E. d e. B f Fn d )
21 fnfun 
 |-  ( f Fn d -> Fun f )
22 20 21 bnj31 
 |-  ( f e. C -> E. d e. B Fun f )
23 22 bnj1265 
 |-  ( f e. C -> Fun f )
24 18 23 bnj593 
 |-  ( f e. H -> E. y Fun f )
25 24 bnj937 
 |-  ( f e. H -> Fun f )